1. By: Amani Albraikan 1

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3.  Synonym for variability  Often called “spread” or “scatter”  Indicator of consistency among a data set  Indicates how close data are clustered about a measure of central tendency  Commonly used measures of dispersion ◦ Range ◦ Interquartile Range ◦ Standard Deviation

4.  Measures of dispersion are descriptive statistics that describe how similar a set of scores are to each other ◦ The more similar the scores are to each other, the lower the measure of dispersion will be ◦ The less similar the scores are to each other, the higher the measure of dispersion will be ◦ In general, the more spread out a distribution is, the larger the measure of dispersion will be 4

5.  Which of the distributions of scores has the larger dispersion? 5 The upper distribution has more dispersion because the scores are more spread out That is, they are less similar to each other

6.  There are three main measures of dispersion: ◦ The range ( المدى ) ◦ The semi-interquartile range (SIR) نصف المدى الربيعي ◦ Variance / standard deviation( التباين و الانحراف المعياري ) 6

7.  The range is defined as the difference between the largest score in the set of data and the smallest score in the set of data, X L - X S  What is the range of the following data: 4 8 1 6 6 2 9 3 6 9  The largest score (X L ) is 9; the smallest score (X S ) is 1; the range is X L - X S = 9 - 1 = 8 7

8.  The range is used when ◦ you have ordinal data or ◦ you are presenting your results to people with little or no knowledge of statistics  The range is rarely used in scientific work as it is fairly insensitive ◦ It depends on only two scores in the set of data, X L and X S ◦ Two very different sets of data can have the same range: 1 1 1 1 9 vs 1 3 5 7 9 8

9.  Used for Interval-Ratio Level Variables  Difference between the Highest and Lowest Values.  Range= Maximum value – minimum value  Often expressed as the full equation (i.e., There was a 19 point range, with a maximum score of 96 and a minimum of 77.) 9

10.  Represents the width of the middle 50% of the data.  The 3 rd quartile (Q3 ; 75 th percentile) minus the 1 st quartile (Q1; 25 th percentile) equals the interquartile range (IQR).  Formula: IQR=Q3-Q1 10

11.  Order the values of an interval-ratio level variable from lowest to highest.  Multiple N by.75 to find where Q3 is located and by.25 to find Q1. If Q3 and Q1 are not even numbers, find the values on either side and average them.  Subtract the value at Q1 from the one at Q3. 11

12.  68, 72, 73, 76, 79, 79, 82, 83, 84, 84, 86, 88, 90, 91, 91, 95, 96, 97, 99, 99  N=20  Q1=20 *.25= 5 th case=79  Q3=20 *.75=15 th case=91  91-79=12  Interpretation: The middle 50% of data falls between 79 and 91, with an inter-quartile range of 12 points. 12

13.  The semi-interquartile range (or SIR) is defined as the difference of the first and third quartiles divided by two ◦ The first quartile is the 25 th percentile ◦ The third quartile is the 75 th percentile  SIR = (Q 3 - Q 1 ) / 2 13

14.  What is the SIR for the data to the right?  25 % of the scores are below 5 ◦ 5 is the first quartile  25 % of the scores are above 25 ◦ 25 is the third quartile  SIR = (Q 3 - Q 1 ) / 2 = (25 - 5) / 2 = 10 14

15.  The SIR is often used with skewed data as it is insensitive to the extreme scores 15

16.  Variance: Used only for interval-ratio level variables. Assesses the average squared deviations of all values from the mean value.  Standard Deviation: The square root of the variance. Roughly estimated to be the average amount of deviation between each score and the mean. 16

17.  Variance for a population is symbolized by:  2  Variance for a sample is s 2 Formula:

18. 18 s 2 is the sample variance, X is a score, X is the sample mean, and N is the number of scores

19.  Calculate the mean  Subtract the mean from each value (x-  x)  Square each deviation from the mean (x-  x) 2  Sum the squared deviations  Divide by N-1

20.  Values= 1, 5, 8, 38, 39, 42  Mean=22  N=6   (x-  x) 2= 1871  1871/ N-1  1871/ 5 = 374.2  S 2 =374.2Value x-  x x-  x 2 11-22=-21441 55-22=-17289 88-22=-14196 3838-22=16256 3939-22=17289 4242-22=20400 1871

21.  Values: ◦ 12, 15, 18, 21, 22, 23, 26, 32

22.  Select “Transform”  Select “Recode into different variables”  Select variable you wish to recode and click on arrow to send it into box on right  Give your new variable a name and a label and click “change”  Select “old and new values”  For each original variable value enter the old value and the new value and click “add”  Decide how you want to handle missing data, etc…  Click “continue”  Click “okay”

23.  Select: Transform, Recode into new variable  Send variable into center box  Create new name and label: “finrecode” & “Opinion of family income 3 categories”  Old values: 1-2=1; 3=2, 4-5=3, missing=missing  Continue  Okay  Add new value labels to new variable: ◦ 1=below average, 2=average, 3=above average

24.  Variance is defined as the average of the square deviations: 24 أو fifi

25.  First, it says to subtract the mean from each of the scores ◦ This difference is called a deviate or a deviation score ◦ The deviate tells us how far a given score is from the typical, or average, score ◦ Thus, the deviate is a measure of dispersion for a given score 25

26.  Variance is the mean of the squared deviation scores  The larger the variance is, the more the scores deviate, on average, away from the mean  The smaller the variance is, the less the scores deviate, on average, from the mean 26

27.  When the deviate scores are squared in variance, their unit of measure is squared as well ◦ E.g. If people’s weights are measured in pounds, then the variance of the weights would be expressed in pounds 2 (or squared pounds)  Since squared units of measure are often awkward to deal with, the square root of variance is often used instead ◦ The standard deviation is the square root of variance 27

28.  Standard deviation =  variance  Variance = standard deviation 2 28

29.  Open Data File: GSS02PFP-B.sav  Calculate the IQV of the variable class  Calculate the Range, IQR, Variance and Standard Deviation of the variables: age, childs, educ, hrs1  Make a box plot with educ and class  Make a box plot with hrs1 and class  Recode hrs1 into hrs2: Convert interval-ratio variable into ordinal variable with 4 categories: 0= 0 Didn’t work, 1-32=1 part-time, 33-45=2 full time, 45 to highest value=3 more than full-time  Make a simple bar graph with the new variable hrs2  Make a clustered bar graph with the new variable hrs2 and another categorical variable (select % instead of count)